Fix non-zero reals α1, … , αn with n≥ 2 and let K be a non-empty open connected set in a topological vector space such that ∑ i≤nαiK⊆ K (which holds, in particular, if K is an open convex cone and α1, … , αn&gt; 0). Let also Y be a vector space over F: = Q(α1, … , αn). We show, among others, that a function f: K→ Y satisfies the general linear equation ∀x1,…,xn∈K,f(∑i≤nαixi)=∑i≤nαif(xi)if and only if there exist a unique F-linear AX→ Y and unique b∈ Y such that f(x) = A(x) + b for all x∈ K, with b= 0 if ∑ i≤nαi≠ 1. The main tool of the proof is a general version of a result Radó and Baker on the existence and uniqueness of extension of the solution on the classical Pexider equation.

### The general linear equation on open connected sets

#### Abstract

Fix non-zero reals α1, … , αn with n≥ 2 and let K be a non-empty open connected set in a topological vector space such that ∑ i≤nαiK⊆ K (which holds, in particular, if K is an open convex cone and α1, … , αn> 0). Let also Y be a vector space over F: = Q(α1, … , αn). We show, among others, that a function f: K→ Y satisfies the general linear equation ∀x1,…,xn∈K,f(∑i≤nαixi)=∑i≤nαif(xi)if and only if there exist a unique F-linear AX→ Y and unique b∈ Y such that f(x) = A(x) + b for all x∈ K, with b= 0 if ∑ i≤nαi≠ 1. The main tool of the proof is a general version of a result Radó and Baker on the existence and uniqueness of extension of the solution on the classical Pexider equation.
##### Scheda breve Scheda completa Scheda completa (DC)
2020
2019
https://www.springer.com/journal/10474/submission-guidelines
Pexider equation; general linear equation; existence and uniqueness of extension; open connected set.
Schwaiger, Jens; Leonetti, P
File in questo prodotto:
File
1905.12023.pdf

accesso aperto

Tipologia: Documento in Pre-print
Licenza: DRM non definito
Dimensione 160.82 kB
s10474-019-00987-6.pdf

non disponibili

Tipologia: Versione Editoriale (PDF)
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11383/2142077`